Journal of Convex Analysis 24 (2017), No. 1, 067--073
Copyright Heldermann Verlag 2017
A Universal Bound on the Variations of Bounded Convex Functions
Institut de Mathématiques, Équipe Combinatoire et Optimisation, Université Pierre-et-Marie-Curie, 4 place Jussieu, 75252 Paris Cedex 05, France
Given a convex set $C$ in a real vector space $E$ and two points $x,y\in C$, we investivate which are the possible values for the variation $f(y)-f(x)$, where $f:C\longrightarrow [m,M]$ is a bounded convex function. We then rewrite the bounds in terms of the Funk weak metric, which will imply that a bounded convex function is Lipschitz-continuous with respect to the Thompson and Hilbert metrics. The bounds are also proved to be optimal. We also exhibit the maximal subdifferential of a bounded convex function at a given point $x\in C$.
Keywords: Convex functions, variations, Funk metric, Thompson metric, Hilbert metric.
MSC: 26B25, 52A05
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